We study a version of the Hermitian curvature flow on compact homogeneous complex manifolds. We prove that the solution has a finite exstinction time T > 0 and we analyze its behaviour when t → T . We also determine the invariant static metrics and we study the convergence of the normalized flow to one of them.

Given a simply connected compact generalized flag manifold M together with its invariant Kähler Einstein metricḡ, we investigate the functional given by the first eigenvalue of the Hodge Laplacian on C ∞ (M ) restricted to the space of invariant Kähler metrics. We give sufficient and necessary conditions so that the metricḡ is a critical point for this functional. Moreover we prove that when M is a full flag manifold, the metricḡ is critical if and only if M = SU(3)/T 2 and in this caseḡ is a maximum.2010 Mathematics Subject Classification. 53C25, 53C21.

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We study the cohomogeneity of possibly reducible representations of copolarity 1 and 2, generalizing results obtained by

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